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Pressure-dipole solutions of the thin-film equation

Published online by Cambridge University Press:  02 April 2018

M. BOWEN  and
T. P. WITELSKI
M. BOWEN
Affiliation:
International Center for Science and Engineering Programs, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan email: bowen@waseda.jp
T. P. WITELSKI
Affiliation:
Department of Mathematics, Duke University, 295 Physics Building, Box 90320, Durham, NC 27708-0320, USA email: witelski@math.duke.edu
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Abstract

We investigate self-similar sign-changing solutions to the thin-film equation, ht = −(|h|nhxxx)x, on the semi-infinite domain x ≥ 0 with zero-pressure-type boundary conditions h = hxx = 0 imposed at the origin. In particular, we identify classes of first- and second-kind compactly supported self-similar solutions (with a free-boundary x = s(t) = Ltβ) and consider how these solutions depend on the mobility exponent n; multiple solutions can exist with the same number of sign changes. For n = 0, we also construct first-kind self-similar solutions on the entire half-line x ≥ 0 and show that they act as limiting cases for sequences of compactly supported solutions in the limit of infinitely many sign changes. In addition, at n = 1, we highlight accumulation point-like behaviour of sign-changes local to the moving interface x = s(t). We conclude with a numerical investigation of solutions to the full time-dependent partial differential equation (based on a non-local regularisation of the mobility coefficient) and discuss the computational results in relation to the self-similar solutions.

Keywords

Thin-film equationSimilarity solutionsSecond-kind solutionsSign-changing solutions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 2 , April 2019 , pp. 358 - 399
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†MB acknowledges the support of a Waseda University Grant for Special Research Projects (Project number: 2017B-161).

References

[1] Álvarez-Caudevilla, P. & Galaktionov, V. A. (2011) Local bifurcation-branching analysis of global and “blow-up” patterns for a fourth-order thin film equation. NoDEA Nonlinear Differ. Eq. Appl. 18 (5), 483–537.Google Scholar
[2] Álvarez-Caudevilla, P. & Galaktionov, V. A. (2015) Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches. Nonlinear Anal. 121, 19–35.Google Scholar
[3] Barenblatt, G. I. (1996) Scaling, Self-similarity, and Intermediate Asymptotics. Cambridge University Press, Cambridge.Google Scholar
[4] Belgacem, F. B., Gnann, M. V. & Kuehn, C. (2016) A dynamical systems approach for the contact-line singularity in thin-film flows. Nonlinear Anal. 144, 204–235.Google Scholar
[5] Beretta, E., Bertsch, M. & Dal Passo, R. (1995) Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation. Arch. Rat. Mech. Anal. 129, 175–200.Google Scholar
[6] Bernis, F. (1988) Source-type solutions of fourth order degenerate parabolic equations. In: Proceedings of the Microprogram on Nonlinear Diffusion Equations and Their Equilibrium States, I, Mathematical Sciences Research Institute Publications, Vol. 12, Berkeley, CA, 1986, Springer, New York, pp. 123–146.Google Scholar
[7] Bernis, F. (1992) Nonlinear parabolic equations arising in semiconductor and viscous droplets models. In: Proceedings of the Program on Nonlinear Diffusion Equations and Their Equilibrium States, III, Gregynog, 1989, Vol. 7, Birkhäuser Boston, Boston, MA, pp. 77–88.Google Scholar
[8] Bernis, F. (1996) Finite speed of propagation and continuity of the interface for thin viscous flows. Adv. Differ. Equ. 1 (3), 337–368.Google Scholar
[9] Bernis, F. & Friedman, A. (1990) Higher order nonlinear degenerate parabolic equations. J. Differ. Equ. 83 (1), 179–206.Google Scholar
[10] Bernis, F., Hulshof, J. & King, J. R. (2000) Dipoles and similarity solutions of the thin film equation in the half-line. Nonlinearity 13 (2), 413–439.Google Scholar
[11] Bernis, F., Hulshof, J. & Quiros, F. (2000) The “linear” limit of thin film flows as an obstacle-type free boundary boundary. SIAM J. Appl. Math. 61 (3), 1062–1079.Google Scholar
[12] Bernis, F., Peletier, L. A., & Williams, S. M. (1992) Source type solutions of a fourth order nonlinear degenerate parabolic equation. Nonlinear Anal. 18, 217–234.Google Scholar
[13] Bernoff, A. J. & Witelski, T. P. (2002) Linear stability of source-type similarity solutions of the thin film equation. Appl. Math. Lett. 15 (5), 599–606.Google Scholar
[14] Bernoff, A. J. & Witelski, T. P. Stability and dynamics of self-similarity in evolution equations. J. Eng. Math. 66 (1–3), 11–31.Google Scholar
[15] Bertozzi, A. L. (1998) The mathematics of moving contact lines in thin liquid films. Notices Amer. Math. Soc. 45 (6), 689–697.Google Scholar
[16] Bertozzi, A. L., Brenner, M. P., Dupont, T. F. & Kadanoff, L. P. (1994) Singularities and similarities in interface flows. Trends Perspect. Appl. Math. 100, 155–208.Google Scholar
[17] Bertozzi, A. L. & Pugh, M. (1996) The lubrication approximation for thin viscous films: Regularity and long-time behaviour of weak solutions. Comm. Pure. Appl. Math. 49, 85–123.Google Scholar
[18] Bowen, M., Hulshof, J. & King, J. R. (2001) Anomalous exponents and dipole solutions for the thin film equation. SIAM J. Appl. Math 62 (1), 149–179.Google Scholar
[19] Bowen, M. & King, J. R. (2001) Asymptotic behaviour of the thin film equation in bounded domains. Eur. J. Appl. Math. 12, 135–157.Google Scholar
[20] Bowen, M. & Witelski, T. P. (2006) The linear limit of the dipole problem for the thin film equation. SIAM J. Appl. Math. 66 (5), 1727–1748.Google Scholar
[21] Budd, C., Dold, B. & Stuart, A. (1993) Blowup in a partial differential equation with conserved first integral. SIAM J. Appl. Math. 53 (3), 718–742.Google Scholar
[22] Cannon, J. R. & Matheson, A. L. (1993) A numerical procedure for diffusion subject to the specification of mass. Int. J. Eng. Sci. 31 (3), 347–355.Google Scholar
[23] Chugunova, M., King, J. R. & Taranets, R. M. (2012) Uniqueness of the regular waiting-time type solution of the thin film equation. Eur. J. Appl. Math. 23 (4), 537–554.Google Scholar
[24] Chugunova, M., Pugh, M. C. & Taranets, R. M. (2010) Nonnegative solutions for a long-wave unstable thin film equation with convection. SIAM J. Math. Anal. 42 (4), 1826–1853.Google Scholar
[25] Constantin, P., Dupont, T. F., Goldstein, R. E., Kadanoff, L. P., Shelley, M. J. & Zhou, S. (1993) Droplet breakup in a model of the Hele-Shaw cell. Phys. Rev. E 47, 4169–4181.Google Scholar
[26] Craster, R. V. & Matar, O. K. (2009) Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131–1198.Google Scholar
[27] Dai, S. & Du, Q. (2016) Weak solutions for the Cahn-Hilliard equation with degenerate mobility. Arch. Ration. Mech. Anal. 219 (3), 1161–1184.Google Scholar
[28] Dziwnik, M., Münch, A. & Wagner, B. (2017) An anisotropic phase-field model for solid-state dewetting and its sharp-interface limit. Nonlinearity 30 (4), 1465.Google Scholar
[29] Elliott, C. M. & Garcke, H. (1996) On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27 (2), 404–423.Google Scholar
[30] Evans, J. D. & Galaktionov, V. A. (2011) On continuous branches of very singular similarity solutions of a stable thin film equation. I—The Cauchy problem. Eur. J. Appl. Math. 22 (3), 217–243.Google Scholar
[31] Evans, J. D. & Galaktionov, V. A. (2011) On continuous branches of very singular similarity solutions of the stable thin film equation. II—Free-boundary problems. Eur. J. Appl. Math. 22 (3), 245–265.Google Scholar
[32] Evans, J. D., Galaktionov, V. A. & King, J. R. (2007) Blow-up similarity solutions of the fourth-order unstable thin film equation. Eur. J. Appl. Math. 18 (2), 195–231.Google Scholar
[33] Evans, J. D., Galaktionov, V. A. & King, J. R. (2007) Source-type solutions of the fourth-order unstable thin film equation. Eur. J. Appl. Math. 18 (3), 273–321.Google Scholar
[34] Fairweather, G. & Saylor, R. D. (1991) The reformulation and numerical solution of certain nonclassical initial-boundary value problems. SIAM J. Sci. Statist. Comput. 12 (1), 127–144.Google Scholar
[35] Giacomelli, L., Gnann, M. V. & Otto, F. (2013) Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3. Eur. J. Appl. Math. 24 (5), 735–760.Google Scholar
[36] Giacomelli, L., Knüpfer, H. and Otto, F. (2008) Smooth zero-contact-angle solutions to a thin-film equation around the steady state. J. Differ. Equ. 245 (6), 1454–1506.Google Scholar
[37] Gnann, M. V. (2015) Well-posedness and self-similar asymptotics for a thin-film equation. SIAM J. Math. Anal. 47 (4), 2868–2902.Google Scholar
[38] Greenspan, H. P. (1978) On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84, 125–143.Google Scholar
[39] Grün, G. (2003) Droplet spreading under weak slippage: a basic result on finite speed of propagation. SIAM J. Math. Anal. 34 (4), 992–1006.Google Scholar
[40] Hocking, L. M. (1981) Sliding and spreading of thin two-dimensional drops. Q. J. Mech. Appl. Math. 34, 37–54.Google Scholar
[41] Hulshof, J., King, J. R. & Bowen, M. (2001) Intermediate asymptotics of the porous medium equation with sign changes. Adv. Diff. Equ. 6 (9), 1115–1152.Google Scholar
[42] Keller, H. B. (1983) The bordering algorithm and path following near singular points of higher nullity. SIAM J. Sci. Statist. Comput. 4 (4), 573–582.Google Scholar
[43] Keller, H. B. (1992) Numerical Methods for Two-Point Boundary Value Problems. Dover Publications Inc., New York.Google Scholar
[44] King, J. R. (1988) Approximate solutions to a nonlinear diffusion equation. J. Eng. Math. 22 (1), 53–72.Google Scholar
[45] King, J. R. (1991) Integral results for nonlinear diffusion equations. J. Eng. Math. 25 (2), 191–205.Google Scholar
[46] King, J. R. (2001) Two generalisations of the thin film equation. Math. Comput. Mod. 34, 737–756.Google Scholar
[47] King, J. R. (2007) Microscale sensitivity in moving-boundary problems for the thin-film equation. In: Control Methods in PDE-dynamical Systems, Contemporary Mathematics., Vol. 426, Amer. Math. Soc., Providence, RI, pp. 269–292.Google Scholar
[48] King, J. R. & Bowen, M. (2001) Moving boundary problems and non-uniqueness for the thin film equation. Eur. J. Appl. Math. 12, 321–356.Google Scholar
[49] King, J. R. & Please, C. P. (1986) Diffusion of dopant in crystalline silicon: an asymptotic analysis. IMA J. Appl. Math. 37 (3), 185–197.Google Scholar
[50] Knüpfer, H. (2015) Well-posedness for a class of thin-film equations with general mobility in the regime of partial wetting. Arch. Ration. Mech. Anal. 218 (2), 1083–1130.Google Scholar
[51] Lin, X.-B. & Schecter, S. (2003) Stability of self-similar solutions of the Dafermos regularization of a system of conservation laws. SIAM J. Math. Anal. 35 (4), 884–921.Google Scholar
[52] Myers, T. G. (1998) Thin films with high surface tension. SIAM Rev. 40 (3), 441–462.Google Scholar
[53] Ockendon, H & Ockendon, J. R. (1995) Viscous Flow. Cambridge University Press.Google Scholar
[54] Oron, A., Davis, S. H. & Bankoff, S. G. (1997) Long-scale evolution of thin liquid films. Rev. Modern Phys. 69 (3), 931.Google Scholar
[55] Schecter, S. (2006) Eigenvalues of self-similar solutions of the Dafermos regularization of a system of conservation laws via geometric singular perturbation theory. J. Dyn. Differ. Equ. 18 (1), 53–101.Google Scholar
[56] Souplet, P. (1998) Blow-up in nonlocal reaction-diffusion equations. SIAM J. Math. Anal. 29 (6), 1301–1334.Google Scholar
[57] Stakgold, I. & Holst, M. (2011) Green's Functions and Boundary Value Problems. John Wiley & Sons, Inc., Hoboken, NJ, third edition.Google Scholar
[58] Taranets, R. M. & King, J. R. (2014) On an unstable thin-film equation in multi-dimensional domains. NoDEA Nonlinear Differ. Equ. Appl. 21 (1), 105–128.Google Scholar
[59] Vázquez, J. L. (2007) The Porous Medium Equation – Mathematical Theory, Oxford Mathematical Monographs. Oxford University Press.Google Scholar
[60] Witelski, T. P., Bernoff, A. J. & Bertozzi, A. L. (2004) Blowup and dissipation in a critical-case unstable thin film equation. European J. Appl. Math. 15 (2), 223–256.Google Scholar
[61] Zhornitskaya, L. & Bertozzi, A. L. Positivity-preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal. 37 (2), 523–555.Google Scholar